Ta có : $S =1.2.3+2.3.4+....+k.(k+1).(k+2) $
$⇒4S = 1.2.3.4+2.3.4.(5-1)+....+k.(k+1).(k+2).(k+3-k+1) $
$⇒4S = 1.2.3.4+2.3.4.5-1.2.3.4+...+k(k+1)(k+2)(k+3)-(k-1).k.(k+1).(k+2)$
$⇒4S = k(k+1)(k+2)(k+3) = (k^2+3k)(k^2+3k+2)$
$⇒4S+1=(k^2+3k).(k^2+3k+2)+1$
$=(k^2+3k)^2+2.(k^2+3k)+1 = (k^2+3k+1)^2$ là số chính phương.